Singular sources in gravity and homotopy in the space of connections

Abstract

Suppose a Lagrangian is constructed from its fields and their derivatives. When the field configuration is a distribution, it is unambiguously defined as the limit of a sequence of smooth fields. The Lagrangian may or may not be a distribution, depending on whether there is some undefined product of distributions. Supposing that the Lagrangian is a distribution, it is unambiguously defined as the limit of a sequence of Lagrangians. But there still remains the question: Is the distributional Lagrangian uniquely defined by the limiting process for the fields themselves? In this paper a general geometrical construction is advanced to address this question. We describe certain types of singularities, not by distribution valued tensors, but by showing that the action functional for the singular fields is (formally) equivalent to another action built out of \emph{smooth} fields. Thus we manage to make the problem of the lack of a derivative disappear from a system which gives differential equations. Certain ideas from homotopy and homology theory turn out to be of central importance in analyzing the problem and clarifying finer aspects of it. The method is applied to general relativity in first order formalism, which gives some interesting insights into distributional geometries in that theory. Then more general gravitational Lagrangians in first order formalism are considered such as Lovelock terms (for which the action principle admits space-times more singular than other higher curvature theories).Comment: 21 pages, 9 figures, RevTe

Generalized phase transitions in Lovelock gravity

“…In the lightlike limit (H → ∞), we get a consistent equation for H either if a → ∞ and H ∼ a d−1 or M + → M − , which coincides with the result of [34]. In the former case, the asymptotic behavior of the potential reads…”

“…To this end, we will split the geometry as M = M − ∪ (Σ × ξ) ∪ M + , where M ∓ denote the interior/exterior regions respectively, Σ is the codimension one junction (hyper)surface, and ξ ∈ [0, 1] is a real segment/parameter used to interpolate both regions. 3 The matching condition can be obtained from an action principle constructed by means of two auxiliary quantities [34]…”

Shock waves and Birkhoff’s theorem in Lovelock gravity

“…The limit we described here as ∆ → Σ, or any limit equivalent to it, was studied for any theory built out of forms fields alone, such as E and ω, in Ref. [50]. The general result is as follows.…”

“…To do this we shall exploit the formulation of matching conditions first presented in Ref. [48] and further elaborated in [49] and [50]. The basic facts of the method can be summarized as follows.…”

“…The way of 'matching' in the formulation of [48][49][50] is somewhat different than usual: One works with globally defined fields with a certain amount of discontinuity at some places and attempts to write down an action functional for these fields which is well defined in the sense stated above. The spirit of the analysis reflects nicely the old work of Papapetrou and Treded [8] in general relativity we already mentioned.…”